In mathematics, an expansion of a product of sums expresses it as a sum of products by using the fact that multiplication distributes over addition. Expansions of polynomials are obtained by multiplying together their factors, which results in a sum of terms with variables raised to different degrees.

Expansion of a polynomial written in factored form

To multiply two factors together, you must multiply every term in both factors by every term in the other factor. If both factors are binomials, the acronym FOIL is used, which stands for First Outer Inner Last, referring the terms that are multiplied together. For example, if you wished to expand $(x+2)(2x-5)$, you would get $2x^2+4x-5x-10$, or $2x^2-x-10$

Expansion of $(x+y)^n$

When expanding $(x+y)^n$, a special relationship exists between the coefficients of the terms when written in order of descending powers of x and ascending powers of y. The coefficients will be the numbers in the (n + 1)-th row of Pascal's triangle.

For example, when expanding $(x+y)^6$, the following is obtained:

$\{\backslash color\{red\}1\}x^6+\{\backslash color\{red\}6\}x^5y+\{\backslash color\{red\}15\}x^4y^2+\{\backslash color\{red\}20\}x^3y^3+\{\backslash color\{red\}15\}x^2y^4+\{\backslash color\{red\}6\}xy^5+\{\backslash color\{red\}1\}y^6\; \backslash ,$

See also

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• Factorization, the opposite of expansion

Polynomials